Second-order fluid

The simplest constitutive equation capable of predicting a first normal stress difference is the equation of the second-order fluid (Bird et al., 1987 Larson, 1988)  

Hereafter in constitutive equations such as eq 4.3.1 we will use the stress tensor r, which does not contain the isotropic pressure term pi. In eq 4.3.1 we have introduced the upper-convected derivative, denoted by V, which when acting on an arbitrary tensor A gives by definition 

As usual, the dot ( ) over a tensor denotes the substantial or material time derivative of that tensor that is. 

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13)  

The term proportional to b in eq 4.3.1 incorporates a weak elastic memory into the constitutive equation. It can be shown under quite general conditions that a viscoelastic fluid will obey eq 4.3.1 if the flow is sufficiently slow and slowly varying, to ensure that 

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Using this equation an attempt was made to find a critical Re-number which could be correlated to the onset of vortices observed with the naked eye, as has been done, for example, for Newtonian fluids [93], but no correlation could be found [88]. The reason is probably due to the fact that polymer solutions are viscoelastic fluids, also called second-order fluids. 

Coleman,B.D., Markovitz,H. Normal stress effects in second-order fluids. J. Appl. Phys. 35,1-9 (1964). 

This behaviour of the extinction angles is in accordance with eq. (1.3), if % = In fact, for a second order fluid the first normal stress difference increases with the square of the shear rate, whereas the shear stress increases with the first power of this rate (constant viscosity). As a consequence, it follows from eq. (1.3) that cot 2% increases linearly. From this fact the above mentioned linear behaviour of the extinction angle curve is deduced, since... 

Fig. 1.4 gives such a plot, which was prepared by Philippoff (8,9) from his early measurements on a 15 per cent solution of polyisobutylene (P-100) in decalin (measurement temperatures 30 and 50°C). From this figure it is clearly seen that An as a function of p21 is non-linear. In contrast to the above mentioned solution of S 111 in methyl 4-bromo-phenyl carbinol, the solution of the poly-isobutylene P-100 in decalin does not form a second order fluid. However, for the product A n sin 2%, one obtains a beautiful straight line. The stress-optical law seems to hold also for this more general type of fluid. ... 

There are some interesting points to be noted. First, it seems that also for polymer melts the normal stress differences (fin — fi22) and (fin—fi33) are practically equal. (Similar results have been obtained for melts of several polyethylenes.) Second, for the investigated polystyrene a practically quadratic dependence of nn — n33 on the shear stress is found up to the point of the inset of an extrusion defect. It is noteworthy that Fig. 1.9 shows no quadratic dependence of Pjd vs Ds, as would be expected for a second order fluid. Third, the measurements in the cone-and-plate apparatus have to be stopped at a shear stress at least one... 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

The course of this curve suggests that the theoretical slope two, which should hold for a second order fluid, will be reached only at shear rates or angular frequencies considerably lower than the ones used in the experiments. It thus appears that the interrelations given by the above mentioned equations hold even outside the literal range of validity of the discussed theory. Similar results were obtained for two samples of linear polyethylene. 

The trouble is that, under transient conditions, the shear recovery vs. preceding shear deformation can be much more sensitive to deviations from the strict behaviour of a second order fluid than the shear viscosity or the normal stress difference. A few entanglements between extraordinarily long chain molecules may be responsible for a maximum in the shear recovery. If this is the case, a shear recovery higher than the one... 

As a final remark it may be mentioned that the discussed polypropylene melts do not at all behave like second-order fluids in the range of shear rates and angular frequencies accessible to measurement. This is shown in Fig. 4.6. In this figure the doubled extinction angle 2 is plotted... 

It is clearly seen that the validity of the stress-optical law is more general than that of the said relation for second order fluids. As the... 

Bueche-Ferry theory describes a very special second order fluid, the above statement means that a validity of this theory can only be expected at shear rates much lower than those, at which the measurements shown in Fig. 4.6 were possible. In fact, the course of the given experimental curves at low shear rates and frequencies is not known precisely enough. It is imaginable that the initial slope of these curves is, at extremely low shear rates or frequencies, still a factor two higher than the one estimated from the present measurements. This would be sufficient to explain the shift factor of Fig. 4.5, where has been calculated with the aid of the measured non-Newtonian viscosity of the melt. A similar argumentation may perhaps be valid with respect to the "too low /efi-values of the high molecular weight polystyrenes (Fig. 4.4). 

Here we have three parameters r/o the zero-shear-rate viscosity, Ai the relaxation time and A2 the retardation time. In the case of A2 = 0 the model reduces to the convected Maxwell model, for Ai = 0 the model simplifies to a second-order fluid with a vanishing second normal stress coefficient [6], and for Ai = A2 the model reduces to a Newtonian fluid with viscosity r/o. If we impose a shear flow,... 

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

For almost steady flows one can expand yl1 or y about t t and obtain second-order fluid constitution equations in the co-deforming frame. When steady shear flows are considered, the CEF equation is obtained, which, in turn, reduces to the GNF equation for T i = 2 = 0 and to a Newtonian equation if, additionally, the viscosity is constant. 

Second-order fluids (3.3-4) Constant vPi are constant and related to each other No No... 

Second-Order Fluids in Simple Shearing Flow 515... 

SECOND-ORDER FLUIDS IN SIMPLE SHEARING FLOW... 

Three functions (q, P, and y) are necessary to characterize the flow of a second-order fluid. Let us determine these functions for a memory fluid under simple shearing flow (25, 26), as shown in Figure 13.1. In this case, the coordinates of a given point at time / will be X =Xi,X2 t) = [x2(/)/xi]xi = Xi tan5(/), and x = x. The velocity field is given by... 

The constitutive equation of second-order fluids indicates that the stress tensor is determined by — u), and consequently Gy t) is independent of t and depends only on and Af Thus the function dy reduces to a function y, and the constitutive equation for second-order fluids [Eq. (13.9)] can be expressed as... 

For simple shear experiments, somewhat lengthy mathematical arguments indicate that the stress tensor for second-order fluids is given by (2)... 

Finally, it should be stressed that second-order fluids present normal stresses and, as a consequence, nonlinear effects at shear rates corresponding to the Newtonian regime. It is noteworthy that the first normal stress coefficient starts at 2Je, and, like the viscosity, it decreases with increasing shear rate for complex fluids. 

L. G. Leal, The slow motion of slender rod-like particles in a second-order fluid, J. Fluid Mech. 69, 305-37 (1975) B. P. Ho and L. G. Leal, Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid, J. Fluid Mech. 76, 783-99 (1976) P. C. H. Chan and L. G. Leal, The motion of a deformable drop in a second-order fluid, J. Fluid Mech. 92, 131-70 (1979) L. G. Leal, The motion of small particles in non-Newtonian fluids, J. Non-Newtonian Fluid Mech. 5, 33-78 (1979) R. J. Phillips, Dynamic simulation of hydro-dynamically interacting spheres in a quiescent second-order fluid, J. Fluid Mech. 315, 345-65 (1996). 

Markovitz, II., Coleman, B. D. Incompressible second-order fluids. Advanc. Appl. Mech. 8, 69-101 (1964). 

Joseph, D. D. and Feng J., A note on the forces that move particles in a second-order fluid, J. Non-Newtonian Fluid Mech., Vol. 64, pp. 299-302,1996. 

Normal stress measurements for some MLC nematics was reported to be consistent with that of a second-order fluid, that the low frequency limit of G /co equaled the low shear limit of N /(dy/dty [36]. Coleman and Markowitz demonstrated that for a second-order fluid in slow Couette flow, the viscoelastic contribution to the normal thrust must have a sign opposite to the inertial contribution on thermodynamic grounds [37]. A textbook by Walters stated that the measurements of first normal stress difference have invariably led to a positive quantity except for one case which was later found to be in error [38]. Adams and Lodge reported the possible observation of a negative value for Nj for solutions of poly isobutylene + decalin [39]. This result was obtained by a combination of obtained from radial... 

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